L11a370

L11a369

L11a371

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Link Presentations

[edit Notes on L11a370's Link Presentations]

Planar diagram presentation	X12,1,13,2 X20,8,21,7 X14,3,15,4 X6,15,7,16 X16,5,17,6 X4,17,5,18 X22,20,11,19 X18,9,19,10 X2,11,3,12 X10,13,1,14 X8,22,9,21
Gauss code	{1, -9, 3, -6, 5, -4, 2, -11, 8, -10}, {9, -1, 10, -3, 4, -5, 6, -8, 7, -2, 11, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{t(2)^2 t(1)^4-t(2) t(1)^4+2 t(2)^3 t(1)^3-5 t(2)^2 t(1)^3+5 t(2) t(1)^3-t(1)^3+t(2)^4 t(1)^2-5 t(2)^3 t(1)^2+9 t(2)^2 t(1)^2-5 t(2) t(1)^2+t(1)^2-t(2)^4 t(1)+5 t(2)^3 t(1)-5 t(2)^2 t(1)+2 t(2) t(1)-t(2)^3+t(2)^2}{t(1)^2 t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{5/2}-4 q^{3/2}+8 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{17}{q^{5/2}}+\frac{15}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{5}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^3+2 a^7 z+a^7 z^{-1} -a^5 z^5-2 a^5 z^3-3 a^5 z-a^5 z^{-1} -2 a^3 z^5-4 a^3 z^3-3 a^3 z-a z^5+z^3 a^{-1} +a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^6 z^{10}-a^4 z^{10}-2 a^7 z^9-6 a^5 z^9-4 a^3 z^9-2 a^8 z^8-3 a^6 z^8-9 a^4 z^8-8 a^2 z^8-a^9 z^7+4 a^7 z^7+11 a^5 z^7-4 a^3 z^7-10 a z^7+8 a^8 z^6+15 a^6 z^6+22 a^4 z^6+7 a^2 z^6-8 z^6+5 a^9 z^5+3 a^7 z^5+a^5 z^5+20 a^3 z^5+13 a z^5-4 z^5 a^{-1} -10 a^8 z^4-14 a^6 z^4-9 a^4 z^4+4 a^2 z^4-z^4 a^{-2} +8 z^4-8 a^9 z^3-7 a^7 z^3-4 a^5 z^3-13 a^3 z^3-6 a z^3+2 z^3 a^{-1} +4 a^8 z^2+5 a^6 z^2-3 a^2 z^2-2 z^2+4 a^9 z-2 a^5 z+3 a^3 z+a z-a^6+a^7 z^{-1} +a^5 z^{-1} }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                       1 -1 4                     3   3 2                   5 1   -4 0                 7 3     4 -2               9 6       -3 -4             8 6         2 -6           7 9           2 -8         6 8             -2 -10       3 7               4 -12     2 6                 -4 -14   1 4                   3 -16   1                     -1 -18 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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